Introduction to Economic Analysis, v. 1.0

5.2 Incidence of Taxes

How much does the quantity fall when a tax is imposed? How much does the buyer’s price rise and the price to the seller fall? The elasticities of supply and demand can be used to answer this question. To do so, we consider a percentage tax t and employ the methodology introduced in Chapter 2 "Supply and Demand", assuming constant elasticity of both demand and supply. Let the equilibrium price to the seller be p_s and the equilibrium price to the buyer be p_b. As before, we will denote the demand function by q_d(p) = ap^-ε and supply function by q_s(p) = bp^η. These prices are distinct because of the tax, and the tax determines the difference:

p_b = (1 + t)p_s.

Equilibrium requires

$$a{p}_d^{-\epsilon }=q_d(p_b)=q_s(p_s)=b{p}_s^{\eta }\text{.}$$

Thus,

$$a{\left((1+t)p_s\right)}^{-\epsilon }=a{p}_d^{-\epsilon }=q_d(p_b)=q_s(p_s)=b{p}_s^{\eta }\text{.}$$

This solves for

$$p_s={\left(\frac{a}{b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}{(1+t)}^{\raisebox{1ex}{$-\epsilon $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}\text{,}$$

and

$$q*=q_s(p_s)=b{p}_s^{\eta }=b{\left(\frac{a}{b}\right)}^{\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}{(1+t)}^{\raisebox{1ex}{$-\epsilon \eta $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}=a^{\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}{b}^{\raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}{(1+t)}^{\raisebox{1ex}{$-\epsilon \eta $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}\text{.}$$

Finally,

$$p_d=(1+t)p_s={\left(\frac{a}{b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}{(1+t)}^{\raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{$\eta +\epsilon $}\right.}\text{.}$$

Recall the approximation ${\left(1+t\right)}^{{}^r}\approx 1+rt\text{.}$

Thus, a small proportional tax increases the price to the buyer by approximately $\frac{\eta t}{\epsilon +\eta }$ and decreases the price to the seller by $\frac{\epsilon t}{\epsilon +\eta }\text{.}$ The quantity falls by approximately $\frac{\eta \epsilon t}{\epsilon +\eta }\text{.}$ Thus, the price effect is mostly on the “relatively inelastic party.” If demand is inelastic, ε is small; then the price decrease to the seller will be small and the price increase to the buyer will be close to the entire tax. Similarly, if demand is very elastic, ε is very large, and the price increase to the buyer will be small and the price decrease to the seller will be close to the entire tax.

We can rewrite the quantity change as $\frac{\eta \epsilon t}{\epsilon +\eta }=\frac{t}{\frac{1}{\epsilon }+\frac{1}{\eta }}\text{.}$ Thus, the effect of a tax on quantity is small if either the demand or the supply is inelastic. To minimize the distortion in quantity, it is useful to impose taxes on goods that either have inelastic demand or inelastic supply.

For example, cigarettes are a product with very inelastic demand and moderately elastic supply. Thus, a tax increase will generally increase the price by almost the entire amount of the tax. In contrast, travel tends to have relatively elastic demand, so taxes on travel—airport, hotel, and rental car taxes—tend not to increase the final prices so much but have large quantity distortions.